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On the Motion of a Small Light Body Immersed in a Two Dimensional Incompressible Perfect Fluid with Vorticity

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Abstract

In this paper we consider the motion of a rigid body immersed in a two dimensional unbounded incompressible perfect fluid with vorticity. We prove that when the body shrinks to a massless pointwise particle with fixed circulation, the “fluid+rigid body” system converges to the vortex-wave system introduced by Marchioro and Pulvirenti (Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96, Springer-Verlag, 1994). This extends both the paper (Glass et al. Bull Soc Math France 142(3):489–536, 2014) where the case of a solid tending to a massive pointwise particle was tackled and the paper (Glass et al. Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid, 2014) where the massless case was considered but in a bounded cavity filled with an irrotational fluid.

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References

  1. Bjorland C.: The vortex-wave equation with a single vortex as the limit of the Euler equation. Commun. Math. Phys. 305(1), 131–151 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Glass O., Lacave C., Sueur F.: On the motion of a small disk immersed in a two dimensional incompressible perfect fluid. Bull. Soc. Math. France 142(3), 489–536 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Glass, O., Munnier, A., Sueur, F.: Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid. (2014, Preprint). arXiv:1402.5387

  4. Glass O., Sueur F.: On the motion of a rigid body in a two-dimensional irregular ideal flow. SIAM J. Math. Anal. 44(5), 3101–3126 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Iftimie D., Lopes Filho M.C., Nussenzveig Lopes H.J.: Two dimensional incompressible ideal flow around a small obstacle. Commun. Partial Diff. Equ. 28(1&2), 349–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kikuchi K.: Exterior problem for the two-dimensional Euler equation. J. Fac. Sci. Univ. Tokyo Sect 1A Math 30(1), 63–92 (1983)

    MathSciNet  MATH  Google Scholar 

  7. Lacave C.: Two-dimensional incompressible ideal flow around a small curve. Commun. Partial Diff. Equ. 37(4), 690–731 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lacave C., Miot E.: Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex. SIAM J. Math. Anal. 41(3), 1138–1163 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lacave, C., Takahashi, T.: Small moving rigid body into a viscous incompressible fluid. (2015, Preprint). arXiv:1506.08964

  10. Lions, P.-L.: Mathematical topics in fluid mechanics. vol. 1, incompressible models. Oxford Lecture Series in Mathematics and its Applications 3 (1996)

  11. Marchioro C., Pulvirenti M.: Vortices and localization in Euler flows. Commun. Math. Phys. 154(1), 49–61 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Marchioro, C., Pulvirenti, M.: Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96. Springer-Verlag, (1994)

  13. Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963) (in Russian). English translation in USSR Comput. Math. & Math. Physics 3, 1407–1456 (1963)

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Authors and Affiliations

  1. CEREMADE, UMR 7534, Université Paris-Dauphine & CNRS, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Olivier Glass

  2. Univ Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ Paris 06, 75013, Paris, France

    Christophe Lacave

  3. Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux, 351 cours de la Libération, 33405, Talence Cedex, France

    Franck Sueur

Authors
  1. Olivier Glass
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  2. Christophe Lacave
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  3. Franck Sueur
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Correspondence to Christophe Lacave.

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Communicated by C. Mouhot

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Glass, O., Lacave, C. & Sueur, F. On the Motion of a Small Light Body Immersed in a Two Dimensional Incompressible Perfect Fluid with Vorticity. Commun. Math. Phys. 341, 1015–1065 (2016). https://doi.org/10.1007/s00220-015-2489-3

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  • DOI: https://doi.org/10.1007/s00220-015-2489-3

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